Formalize Open Quantum Problem #26: Bell inequalities holding for all quantum states

### What is the conjecture

This is problem [#26](https://oqp.iqoqi.oeaw.ac.at/bell-inequalities-holding-for-all-quantum-states) in [Reinhard F. Werner’s collection](https://arxiv.org/abs/quant-ph/0504166), later [collected by a community of quantum researchers](https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems).

> **Problem (Open Quantum Problem #26: “Bell inequalities holding for all quantum states”).**
> In an `(N,M,K)` Bell scenario, compare the local polytope `C`, the quantum correlation set `Q`, and the no-signalling polytope `P`.
> Problem 26.A asks whether every genuinely quantum boundary point of `Q` can already be realized using pure states, local `K`-dimensional Hilbert spaces, and complete `K`-outcome projective measurements.
> Problem 26.B asks whether every nontrivial facet Bell inequality for `C` is violated by some quantum correlation, or equivalently whether `C` and `Q` can share any facet that is not already inherited from `P`.

A standard formalization is in terms of **Bell scenarios** and the nested sets
`C_{N,M,K} ⊆ Q_{N,M,K} ⊆ P_{N,M,K}`:

* Fix integers `N,M,K ≥ 2`.
* A **behavior** is a family of conditional probabilities
`p(a_1,...,a_N | x_1,...,x_N)`,
where `x_i ∈ {1,...,M}` is the measurement choice of party `i` and `a_i ∈ {1,...,K}` is its outcome, satisfying
`p(a|x) ≥ 0` and `∑_a p(a|x) = 1` for every `x`.
* `p` is **no-signalling** iff every marginal distribution for a subset of parties depends only on the measurement choices of those parties.
The set of all such behaviors is the **no-signalling polytope** `P_{N,M,K}`.
* `p` is **local / classical** iff it admits a local hidden-variable decomposition
`p(a|x) = ∑_λ q(λ) ∏_{i=1}^N p_i(a_i | x_i, λ)`.
Equivalently, it lies in the convex hull of deterministic local strategies. This is the **local polytope** `C_{N,M,K}`.
* `p` is **quantum** iff there exist local Hilbert spaces `H_1,...,H_N`, a state `ρ` on `H_1 ⊗ ··· ⊗ H_N`, and local POVMs `{E_i^{x_i}(a_i)}` such that
`p(a|x) = Tr[ρ (E_1^{x_1}(a_1) ⊗ ··· ⊗ E_N^{x_N}(a_N))]`.
The set of all such behaviors is the **quantum set** `Q_{N,M,K}`.

A **Bell inequality** is a linear functional
`B(p) = ∑_{a,x} β_{a,x} p(a|x) ≤ L`.
It is **tight** / **proper** if it defines a facet of `C_{N,M,K}` and is not merely a trivial inequality coming from positivity, normalization, or no-signalling (equivalently, the corresponding face of `C` is not already a face of `P`).

The title phrase **“holding for all quantum states”** can then be formalized as:
for every quantum behavior `q ∈ Q_{N,M,K}`, one has `B(q) ≤ L`.

With this notation, Problem 26.B becomes:

* **Main question / open geometric formulation:** for every proper Bell inequality `B(p) ≤ L` for `C_{N,M,K}`, does there exist a quantum behavior `q ∈ Q_{N,M,K}` with `B(q) > L`?
* Equivalently, do `C_{N,M,K}` and `Q_{N,M,K}` share any nontrivial facets/supporting hyperplanes beyond those inherited from `P_{N,M,K}`?
* Equivalently again, does there exist a proper Bell inequality with
`sup_{q ∈ Q_{N,M,K}} B(q) = L`,
i.e. a nontrivial Bell inequality that is satisfied by all quantum behaviors?

For Problem 26.A, one natural formalization is to define `Q_{N,M,K}^{min}` as the set of behaviors realizable by pure states on local `K`-dimensional Hilbert spaces using complete `K`-outcome projective measurements, and ask whether every genuinely quantum boundary behavior already lies in `Q_{N,M,K}^{min}`.

The current status is that the two sub-problems behave differently:

* **Problem 26.A** has a **negative** answer in general.
* **Problem 26.B** has a **negative** answer for some **multipartite** scenarios (`N ≥ 3`), but remains open in its natural **bipartite** form.
* So the central remaining open version is:
determine whether every proper **bipartite** Bell inequality is violated by some bipartite quantum correlation.

### Where to find the details / references

Primary sources:

* [Open Quantum Problems site (Problem #26)](https://oqp.iqoqi.oeaw.ac.at/bell-inequalities-holding-for-all-quantum-states)
* [Open Quantum Problems master list (for numbering/metadata)](https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems)
* O. Krüger & R. F. Werner, “Some Open Problems in Quantum Information Theory” (Problem 26):
[https://arxiv.org/abs/quant-ph/0504166](https://arxiv.org/abs/quant-ph/0504166)
[https://arxiv.org/pdf/quant-ph/0504166](https://arxiv.org/pdf/quant-ph/0504166)

Key related references:

* Ll. Masanes, “Extremal quantum correlations for `N` parties with two dichotomic observables per site” (arXiv: quant-ph/0512100)
(proves Problem 26.A in the special case `(N,M,K) = (N,2,2)`)
* T. Vértesi and K. F. Pál, “Generalized Clauser-Horne-Shimony-Holt inequalities maximally violated by higher dimensional systems” (Phys. Rev. A 77, 042106 (2008); arXiv: 0712.4225)
(negative answer to Problem 26.A in general)
* M. L. Almeida, J.-D. Bancal, N. Brunner, A. Acín, N. Gisin, and S. Pironio, “Guess your neighbour’s input: a multipartite non-local game with no quantum advantage” (Phys. Rev. Lett. 104, 230404 (2010); arXiv: 1003.3844)
(negative answer to Problem 26.B for more than two parties)
* N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality” (Rev. Mod. Phys. 86, 419 (2014); arXiv: 1303.2849)
(standard review/background, including the bipartite facet question)
* R. Ramanathan, M. T. Quintino, A. B. Sainz, G. Murta, and R. Augusiak, “On the tightness of correlation inequalities with no quantum violation” (Phys. Rev. A 95, 012139 (2017); arXiv: 1607.05714)
(shows important bipartite no-quantum-advantage correlation inequalities are not facets)
* W. Slofstra, “The set of quantum correlations is not closed” (Forum Math. Pi 7, e1 (2019); arXiv: 1703.08618)
(shows finite input/output Bell scenarios can require limiting / infinite-dimensional quantum strategies)
* L. Escolà-Farràs, J. Calsamiglia, and A. Winter, “All tight correlation Bell inequalities have quantum violations” (Phys. Rev. Research 2, 012044 (2020); arXiv: 1908.06669)
(settles the two-party XOR / correlation-facet case positively)
* R. Ramanathan, “All two-party facet Bell inequalities are violated by Almost Quantum correlations” (Phys. Rev. Research 3, 033100 (2021); arXiv: 2004.07673)
(shows the stronger “almost-quantum” relaxation violates every two-party facet Bell inequality)
* T. P. Le, C. Meroni, B. Sturmfels, R. F. Werner, and T. Ziegler, “Quantum Correlations in the Minimal Scenario” (Quantum 7, 947 (2023); arXiv: 2111.06270)
(modern detailed study of the smallest bipartite scenario `(2,2,2)`)

### Prerequisites needed

* Finite probability distributions and conditional probability tables
* Bell scenarios: local hidden-variable models, no-signalling, and quantum behaviors
* Basic convex geometry: convex hulls, polytopes, faces/facets, supporting hyperplanes
* Basic linear algebra / quantum mechanics: Hilbert spaces, tensor products, density operators, POVMs, projective measurements
* (Optional, for partial results and computational approaches) semidefinite programming, Tsirelson bounds, XOR games / nonlocal games, and dimension witnesses

### [AMS categories](https://github.com/google-deepmind/formal-conjectures/labels?q=ams-)

* ams-81 (Quantum theory)
* ams-52 (Convex and discrete geometry)
* ams-68 (Computer science)
* ams-94 (Information and communication theory)

### Choose either option

* [ ] I plan on adding this conjecture to the repository
* [x] This issue is up for grabs: I would like to see this conjecture added by somebody else

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